Basic Properties of the Poisson Process

Abstract

The archetypal point processes are the Poisson and renewal processes. Their importance is so great, not only historically but also in illustrating and motivating more general results, that we prefer to give an account of some of their more elementary properties in this and the next two chapters before proceeding to more complex examples and the general theory of point processes. For our present purposes, we shall understand by a point process some method of randomly allocating points to intervals of the real line or (occa-sionally) to rectangles or hyper-rectangles in a d-dimensional Euclidean space R d. It is intuitively clear and will be made rigorous in Chapters 5 and 9 that a point process is completely defined if the joint probability distributions are known for the number of events in all finite families of disjoint intervals (or rectangles, etc.). We call these joint or finite-dimensional distributions fidi distributions for short. 2.1. The Stationary Poisson Process With the understanding just enunciated, the stationary Poisson process on the line is completely defined by the following equation, in which we use N (a i , b i ] to denote the number of events of the process falling in the half-open interval (a i , b i ] with a i < b i ≤ a i+1 : Pr{N (a i , b i ] = n i , i = 1,. .. , k} = k i=1 [λ(b i − a i)] ni n i ! e −λ(bi−ai). (2.1.1) This definition embodies three important features: (i) the number of points in each finite interval (a i , b i ] has a Poisson distribution ; 19